Integrable highest weight modules: the weight system, the contravariant Hermitian form and the restriction problem

  • Victor G. Kac
Part of the Progress in Mathematics book series (PM, volume 44)


In this chapter we describe in detail the weight system of an integrable highest weight module L(Λ) over a Kac-Moody algebra g(A). We establish the existence of a Ii(A)-invariant positive-definite Hermitian form on L(Λ). Finally, we study the decomposition of L(Λ) with respect to various subalgebras of g(A) and derive an explicit description of the region of convergence of ch L (Λ).


Convex Hull Finite Type Inductive Assumption Hermitian Form Weight System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical notes and comments

  1. Kac, V. G., Peterson, D. H. [1983 A] Infinite dimensional Lie algebras, theta functions and modular forms, Advances in Math., 50 (1983).Google Scholar
  2. Kac, V. G., Peterson, D. H:[1983 C] Unitary structure in representations of infinite-dimensional groups and a convexity theorem, MIT, preprintGoogle Scholar
  3. Bourbaki, N:[1975] Groupes et algèbres de Lie, Ch. 7–8, Hermann, Paris, 1975.Google Scholar
  4. Garland, H. [1978] The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480–551.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Feingold, A. J., Lepowsky, J. [1978] The Weyl-Kac character formula and power series identities, Adv. Math. 29 (1978), 271–309.MathSciNetzbMATHGoogle Scholar
  6. Kac, V. G., Peterson, D. H. [1983 B] Regular functions on certain infinite dimensional groups, in Arithmetic and Geometry, (ed. M. Artin and J. Tate ), 141–166, Birkhäuser, Boston, 1983.Google Scholar
  7. Berman, S., Moody, R. V. [1979] Multiplicities in Lie algebras, Proc. Amer. Math. Soc. 76 (1979), 223–228.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Feingold, A. J., Frenkel, I. B. [1983] A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2, Math. Ann. 263, (1983), 87–144.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Victor G. Kac
    • 1
  1. 1.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations